# Properties

 Label 4-389-1.1-c1e2-0-0 Degree $4$ Conductor $389$ Sign $1$ Analytic cond. $0.0248029$ Root an. cond. $0.396849$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s − 4-s + 5-s + 2·6-s − 3·7-s + 8-s + 2·9-s − 10-s + 4·11-s + 2·12-s + 13-s + 3·14-s − 2·15-s − 16-s − 2·18-s − 19-s − 20-s + 6·21-s − 4·22-s − 2·24-s + 25-s − 26-s − 6·27-s + 3·28-s + 2·29-s + 2·30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 2/3·9-s − 0.316·10-s + 1.20·11-s + 0.577·12-s + 0.277·13-s + 0.801·14-s − 0.516·15-s − 1/4·16-s − 0.471·18-s − 0.229·19-s − 0.223·20-s + 1.30·21-s − 0.852·22-s − 0.408·24-s + 1/5·25-s − 0.196·26-s − 1.15·27-s + 0.566·28-s + 0.371·29-s + 0.365·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$389$$ Sign: $1$ Analytic conductor: $$0.0248029$$ Root analytic conductor: $$0.396849$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 389,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1979862013$$ $$L(\frac12)$$ $$\approx$$ $$0.1979862013$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad389$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 10 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2^2$ $$1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 - T - p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )$$
17$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$D_{4}$ $$1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$D_{4}$ $$1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$D_{4}$ $$1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$D_{4}$ $$1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 9 T + 118 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 18 T + p T^{2} )$$
97$D_{4}$ $$1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$